Question: A set of biology exam scores are normally distributed with a mean of $70$ points and a standard deviation of $6$ points. Let $X$ represent the score on a randomly selected exam from this set. Find $P(64<X<76)$. You may round your answer to two decimal places.
Explanation: Representing probability with area Since we know the distribution of scores is normally distributed, the probability $P(64<X<76)$ can be found by calculating the shaded area between $X=64$ and $X=76$ in the corresponding normal distribution: $52$ $58$ $64$ $70$ $76$ $82$ $88$ $ \mu_X = 70$ $ \sigma_X = 6$ $ P(64<X<76)$ Calculating shaded area We can use the "normalcdf" function on most graphing calculators to find the shaded area: $\begin{aligned} &\text{normalcdf:} \\\\ &\text{lower bound: } 64 \\\\ &\text{upper bound: } 76 \\\\ &\mu=70 \\\\ &\sigma=6 \end{aligned}$ Output: $\approx0.682689$ [Why do we use normalcdf instead of normalpdf?] Answer $P(64<X<76)\approx0.68$ [Is there another way?]